ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  reuun2 Structured version   GIF version

Theorem reuun2 3214
Description: Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
Assertion
Ref Expression
reuun2 x B φ → (∃!x (AB)φ∃!x A φ))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem reuun2
StepHypRef Expression
1 df-rex 2306 . . 3 (x B φx(x B φ))
2 euor2 1955 . . 3 x(x B φ) → (∃!x((x B φ) (x A φ)) ↔ ∃!x(x A φ)))
31, 2sylnbi 602 . 2 x B φ → (∃!x((x B φ) (x A φ)) ↔ ∃!x(x A φ)))
4 df-reu 2307 . . 3 (∃!x (AB)φ∃!x(x (AB) φ))
5 elun 3078 . . . . . 6 (x (AB) ↔ (x A x B))
65anbi1i 431 . . . . 5 ((x (AB) φ) ↔ ((x A x B) φ))
7 andir 731 . . . . . 6 (((x A x B) φ) ↔ ((x A φ) (x B φ)))
8 orcom 646 . . . . . 6 (((x A φ) (x B φ)) ↔ ((x B φ) (x A φ)))
97, 8bitri 173 . . . . 5 (((x A x B) φ) ↔ ((x B φ) (x A φ)))
106, 9bitri 173 . . . 4 ((x (AB) φ) ↔ ((x B φ) (x A φ)))
1110eubii 1906 . . 3 (∃!x(x (AB) φ) ↔ ∃!x((x B φ) (x A φ)))
124, 11bitri 173 . 2 (∃!x (AB)φ∃!x((x B φ) (x A φ)))
13 df-reu 2307 . 2 (∃!x A φ∃!x(x A φ))
143, 12, 133bitr4g 212 1 x B φ → (∃!x (AB)φ∃!x A φ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628  wex 1378   wcel 1390  ∃!weu 1897  wrex 2301  ∃!wreu 2302  cun 2909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-reu 2307  df-v 2553  df-un 2916
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator